.Got a question here....
Find the derivative of :
P(t) = 500 / (1 + e^-t)
I started by re-writing it in the form :
P(t) = 500 (1 + e^-t)^-1
Using the chain rule I came up with :
P'(t) = 500(-1)(1 + e^-t)^-2 (1 + e^-t)'
= -500(1 + e^-t)^-2(e^-t)(ln e)
From where did you bring that ? It can only confuse you.
= -500(1 + e^-t)^-2(e^-t)(1)
The question goes on to ask for the derivative at P'(3)
P'(3) = -500(1 + e^-3)^-2(e^-3)
Here's the mistake: you actually have , which is positive indeed.
In fact now I realized that that -2 is a power (!!), but still: you wrote , which is lacking a minus sign there!
I know the answer for this question should not be negative as it deals with population regrowth. Where did I go wrong? Thanks.